Exhaust Shakes

Ok so I've been hunting for this shaking/rattling noise for over a year now. I've replaced a bunch of stuff on it and have done some tests on it. Did a compression test and tested fine. Replaced a front engine mount. Replaced engine filter and PCV valve. Ripped off the rotted heat shield that was covering the catalytic converter (that was the first thing I checked). Today after I replaced an outer tie rod end, I decided to try again to figure out and try to narrow down what's causing the exhaust to shake violently especially when the A/C is on. Keep in mind, the A/C has to be engaged and the shaking/rattling only starts when the cooling fan kicks on and then stops when the fan shuts off in an endless cycle until you shut the A/C off. While I had the car on the lift, I took a pry bar and was trying different angles and such to hold the exhaust with the pry bar. I only had the small spurts of time the fan would kick on to try this. I noticed right away that if I pushed the entire exhaust forward towards the front of the vehicle, the shaking/rattling stopped. When I let go, it starts back up. Even if I push the exhaust forward with my hands (while holding rags) and not the pry bar, I get the same result. Can anyone help me from this point on? Its a factory exhaust and has never been taken off.